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WMA11/01 Jan 2022, Question 3: Solving Equations with Surds

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Expand and simplify a surd expression into a specified quadratic form, then solve a linear equation involving surds and give the answer in a required surd form.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (i) · 3 marks

Expand the two squared brackets and collect terms to express f(x) as ax² + bx√2 + c, finding integers a, b and c.

  1. 1√8 = 2√2, so 3x − 5√8 = 3x − 10√2
  2. 2(x + √2)² = x² + 2x√2 + 2
  3. 3(3x − 10√2)² = 9x² − 60x√2 + 200
  4. 4f(x) = 10x² − 58x√2 + 202

Answerf(x) = 10x² − 58x√2 + 202

Part (ii) · 4 marks

Solve the surd equation for y, giving the answer as p + q√3 with p and q as simplified fractions.

  1. 1Expand: 4√3 y − 9 = 5y + √3
  2. 2Collect y terms: (4√3 − 5)y = 9 + √3
  3. 3y = (9 + √3)/(4√3 − 5)
  4. 4Multiply top and bottom by (4√3 + 5): y = (9 + √3)(4√3 + 5)/(48 − 25)
  5. 5Numerator: 36√3 + 45 + 12 + 5√3 = 57 + 41√3
  6. 6y = (57 + 41√3)/23 = 57/23 + (41/23)√3

Answery = 57/23 + (41/23)√3

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